Natural frequencies and mode shapes of uniform cantilever beams are obtained with use of the first and second order central difference schemes. It is observed that the improved finite difference scheme with second order central differences produces the natural frequencies and characteristic functions with a rapid convergence as compared to the conventional approach of using the first order central differences. Further, the present approach facilitates a direct determination of the dynamic characteristics of beams without any necessity of extrapolations of the results or application of iterative procedures for improving the accuracy. Satisfaction of the boundary conditions and elimination of the fictitious stations lying outside the beam domain has been a serious problem in the previous investigations in which higher order central differences have been used. In the present study some recursive relations have been implemented which can be derived by using the first order central differences and application of these simple and elegant relations to higher order difference schemes has been demonstrated to improve the accuracy over conventional approaches. © 1985.
Subrahmanyam, K. B., & Leissa, A. W. (1985). An improved finite difference analysis of uncoupled vibrations of cantilevered beams. Journal of Sound and Vibration, 98(1), 1–11. https://doi.org/10.1016/0022-460X(85)90397-9